Loop Equations and Virasoro Constraints in Matrix Models

Abstract

In the first part of the talk, I review the applications of loop equations to the matrix models and to 2-dimensional quantum gravity which is defined as their continuum limit. The results concerning multi-loop correlators for low genera and the Virasoro invariance are discussed. The second part is devoted to the Kontsevich matrix model which is equivalent to 2-dimensional topological gravity. I review the Schwinger--Dyson equations for the Kontsevich model as well as their explicit solution in genus zero. The relation between the Kontsevich model and the continuum limit of the hermitean one-matrix model is discussed.

22 pages. Talk at XXVth International Symposium on the Theory of Elementary Particles, Gösen, Germany, September 23--26

References in corpus (1)

• On Equivalence of Topological and Quantum 2d Gravity

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• The Schwinger Dyson equations and the algebra of constraints of random tensor models at all orders
• Invitation to Random Tensors
• The Double Scaling Limit in Arbitrary Dimensions: A Toy Model
• The Multicritical Kontsevich-Penner Model
• Abstract loop equations, topological recursion, and applications
• Hopf Algebraic Structure for Tagged Graphs and Topological Recursion